It is oftentimes desirable to determine the value of a contingent claim that may be exercised at some time in the future. The two most common forms of a contingent claim are a call and a put, both of which may arise in a wide variety of applications. For example, financial options commonly involve a call in which a stock or other financial instrument may be purchased at some time in the future for a predetermined exercise price or a put in which a stock or other financial instrument may be sold at some time in the future for a predetermined exercise price. While contingent claims frequently occur in the financial arena, contingent claims also arise in a number of other contexts, such as project evaluation and the evaluation of options to purchase or sell other assets, as described below. Unfortunately, the contingent claims that arise in these other contexts may be more difficult to evaluate than the contingent claims that arise in the financial context since the underlying assets in these other contexts are not traded or valued by a well established market, such as the stock market in the financial arena.
By way of example of the contingent claims that occur in contexts other than the financial arena, the contingent claims that arise during project evaluation and options to purchase or sell other assets will be hereinafter described. In this regard, a number of projects are structured so as to include a contingent claim that may be exercised by one of the participants at some time in the future. The contingent claim oftentimes comes in the form of a call in which one of the participants has an option to invest additional amounts of money in order to continue the project. As such, if the initial stages of the project have proved unsuccessful and/or if the future prospects for the project appear bleak, the participant capable of exercising the call will likely decline to invest additional money and thereby forego exercise of the call and will therefore terminate its participation in the project. Alternatively, if the initial stages of the project have been successful and/or if the prospects of success of the project are bright, the participant capable of exercising the call will likely make the necessary investment in order to continue its participation in the project.
Examples of projects that include a contingent claim at some subsequent time are widely varied, but one common example involves a project having a pilot phase extending from some initial time to a subsequent time at which the contingent claim may be exercised. If the contingent claim is exercised, such as by one of the participants contributing the necessary investment to the project, the project will enter a commercial phase. As a more specific example, the project may involve research and development having staged investments in which each investment is essentially a contingent claim with the participant opting to continue with the research and development activity if the participant makes the necessary investment, but withdrawing from the research and development activity if the participant declines to make the investment. By way of other specific examples, the contingent claim may represent an option for the participant to adjust its production level at a subsequent point in time or an option to adjust its production mix in the future.
In addition to project analysis, contingent claims may arise in the context of an option to purchase or sell assets other than financial assets. In such contexts, the contingent claim oftentimes comes in the form of a call or a put in which one of the participants purchases the contingent claim to thereby have an option to purchase an asset or sell an asset at some subsequent time for a predetermined exercise price. The asset in such contexts can comprise any of a number of different assets, both tangible and intangible assets, including goods, services, and licenses such as cruise ship tickets, tickets to the theatre or a sporting event, the rental of a hotel room, and the rental of a car. In a more specific example, then, the contingent claim may comprise an option to purchase an airline ticket with the option being purchased at some initial time, and the option capable of being exercised at a subsequent time to purchase the airline ticket.
In another similar example, the contingent claim may comprise an option to obtain a full refund on an asset purchased at some initial time, with the option being exercisable at a subsequent time to obtain a full refund. In a more specific example, the asset may comprise an airline ticket purchased at some initial time, where the airline ticket is purchased with an option to obtain a refund of the purchase price at a subsequent time at which the option may be exercised. If the option, or contingent claim, is exercised, the purchaser will then be able to obtain a refund of the purchase price of the ticket by selling the ticket back to the airline ticket vendor (e.g., airline).
Regardless of the type of contingent claim, it is desirable to determine the value of a project and, in particular, the contingent claim at the present time. By determining the value of the contingent claim, the participant can avoid overpaying for the project or asset as a result of an overvaluation of the contingent claim. Conversely, the participant can identify projects or assets in which the value of the contingent claim has been undervalued and can give strong consideration to investing in these projects or assets since they likely represent worthwhile investment opportunities.
Several techniques have been utilized to determine the value of a project or an asset having a contingent claim at a subsequent time. By way of example, techniques utilized in the context of project evaluation will be hereinafter described. For reasons set forth below, each of these techniques has had difficulty evaluating projects involving real options, that is, contingent claims in assets or activities as opposed to financial assets; in large part since assets and activities are not traded in an organized market in the same manner as financial assets.
One technique that has been utilized is the net present value (NPV) method which generally understates the value of a project by ignoring the option to terminate the project at a subsequent time in order to avoid additional investment in a financially unattractive project. A second technique is a decision tree method which does account for the ability to terminate the project at a subsequent time in order to avoid further investment in a financially unattractive project, but which utilizes an incorrect discount rate. In this regard, the decision tree method does not utilize a discount rate that reflects the underlying risks associated with the contingent claim and, as such, generally overstates the value of the project.
Another technique is the Black-Scholes method, which is widely utilized for option pricing in financial markets. The Black-Scholes method can be expressed algorithmically as follows:C0=S0N(d1)−Xe−rtN(d2)whereind1=ln(S0/X)+(r+σ2/2)T d2=d1−σ√{square root over (T)}and wherein C0=f(S0,X,T,r,σ) in which S0 is the value of the project without the real option, X is the contingent investment, T is the duration of the pilot project, i.e., that period of the project that precedes the contingent investment, r is the continuously compounded, risk-free rate of interest, and σ is a volatility parameter.
Of these parameters, S0 and σ can be estimated using the following formulae using the mean E and the variance Var of the project value at the end of the pilot project, i.e., at the subsequent time at which the contingent claim is to be exercised, in which the mean E and the variance Var are defined as follows:E(ST)=S0eμT Var(ST)=S02e2μT(eσ2T−1)
By utilizing the Black-Scholes method, the value of a project having a contingent claim or call option at a subsequent time can be properly valued so long as the various assumptions upon which the Black-Scholes formula is premised hold true. In this regard, the Black-Scholes model assumes, among other things, that the distribution of contingent future benefits is a lognormal distribution. While a lognormal distribution is reasonable for the evaluation of most financial options, the distribution of contingent future benefits attributable to the exercise of a real option during the course of a project may have other types of distributions such that the valuation of a project according to the Black-Scholes method in these instances may be inaccurate. For example, real options, as well as financial options, can periodically experience large amounts of uncertainty such that the value of contingent future benefits experiences rapid changes, or jumps, at various points over time. Such a rapidly changing value of contingent future benefits is sometimes referred to as the jump-diffusion model of future benefits. With the contingent future benefits experiencing rapid changes, then, the distribution of future benefits is not a lognormal distribution, as is assumed by the Black-Scholes model, but instead is defined by the jump-diffusion model.
Additionally, the Black-Scholes formula presumes that the exercise of a contingent claim involves the investment of a predetermined amount of money at a single time in the future. However, some contingent claims are structured to have two or more points in time in which a participant must separately decide whether to pay additional money in order to exercise respective options. Moreover, some contingent claims are structured such that the exercise price to be paid at some time in the future is not a single predetermined amount of money, but rather is best represented by a distribution of investment levels and respective probabilities.
Furthermore, the Black-Scholes formula presumes that the potential loss at the time of exercising the contingent claim is zero since an investor will not exercise an option which will be financially unattractive. For example, if an investor has an option to purchase a stock at a future exercise time for $10, the Black-Scholes formula presumes that the investor will not exercise the option if, at the future exercise time, the stock is selling for less than $10. In contrast, the exercise of contingent claims in other contexts, such as project valuation, is oftentimes not as simple and may still include a potential loss at the time of exercising the contingent claim.
For each of the foregoing reasons, the Black-Scholes formula may therefore be inapplicable to the evaluation of contingent claims in at least some contexts outside of the financial arena. In this regard, contingent claims involving real options may not be properly evaluated by the Black-Scholes formula since the various assumptions upon which the Black-Scholes formula is premised may not hold true.
Additionally, one feature of the Black-Scholes model that was instrumental in its widespread adoption in the context of the valuation of financial options actually renders the Black-Scholes model somewhat difficult to utilize in the context of the valuation of real options. In this regard, the parameters that are utilized in order to value a financial option by means of the Black-Scholes model are relatively intuitive in the financial context. However, the application of the Black-Scholes model to the valuation of a project having a contingent claim, i.e., a real option, becomes problematic since the parameters that are utilized by the Black-Scholes model do not arise naturally in a traditional project analysis. For example, the volatility parameter required by the Black-Scholes model is not commonly utilized during the project analysis. In order to employ the Black-Scholes method, the parameters that arise naturally in the project context must be translated into the parameters that are utilized by the Black-Scholes method. This translation may become a convoluted exercise, and the process quickly loses its intuitive interpretation. Without this intuitive interpretation, project analysts may place less weight or reliance upon the value of a project determined according to the Black-Scholes model since these project analysts may not have a reasonable understanding of the methodology utilized by the Black-Scholes model. As such, the planning and auditing of the valuation of a project is generally more difficult without this intuitive interpretation.
For purposes of comparison of the traditional techniques for the valuation of a project, consider a project having the following parameters:
Mean of ST=$4,375 million
Std Dev of ST=$1,345 million
T=5 years
r=5.5% continuous
weighted average cost of capital (WACC)=10.5% continuous, and
X=$5,000 million
Based upon these parameters, the NPV method would underestimate the project value to be −$370 million, while the decision tree method would overestimate the project value to be $178 million. While the Black-Scholes model correctly determines the project value to be $45 million, the Black-Scholes model is somewhat difficult to utilize since the parameters cannot be directly utilized, but must first be translated as described above, thereby quickly robbing the Black-Scholes model of its intuitive interpretation. As also described above, the Black-Scholes model will generally only provide an accurate valuation of a project so long as the project adheres to all of the assumptions upon which the Black-Scholes model is premised, thereby limiting the applicability of the Black-Scholes model for purposes of contingent claims valuation outside of the financial arena.